Wireframes with Square Cross-Sections?
Some dirty ways

1) Inset faces

Here's a possible workaround based on Inset faces (Individually), Dissolve edges and Extrude (Vertex normals) . It won't give you perfect results, but maybe can help.

First step is select all, than press I twice and tune the inset amount.



Switch to vertex selection and invert selection. Store the selection in a vertex group. Invert again and delete the faces. Than select the items from previously created vertex group and run Dissolve Edges command.



At this point we should have obtain a pletora of n-gons, most of them will probably be non flat and have other issues, but they should approximate our mesh not too badly.

Extrude the Region along Vertex Normals by an amount that may do the section look like if it is a square (or a rectangle) and you'll get the illusion of a squared section wireframe.

Result:



This method works best when there is a smooth transition in surface's curvature. For the 90° degrees angles of the image below it was needed to fix the n-gons (with Split Non Planar Faces command) as their non-planar nature was too pronounced:

 

Notice how the polygon below the gizmo is clearly distorted.



2) Bevel Wireframe modifier

Another possible workaround could be to take advance of the wireframe modifier:

Add and apply a Wireframe modifier:



Select one vertex of the four that marks each hole and then extend selection to all the vertices that has the same amount of connecting edges



Invert selection and then Bevel the edges connected by the freshly selected vertices 



Invert face selection, delete the faces and then delete one of the two resulting mesh pieces too.



Finish by extruding the faces with a Solidify modifier with an appropriate value of Thickness (to match the beveled edge lenght).

Final result: